Detection of number of components in CANDECOMP/PARAFAC models via minimum description length

• The problem of CANDECOMP/PARAFAC (CP) model order selection is addressed.
• Computational efficiency is achieved via matricization of a tensor.
• Eigenvalues associated with the block or multi-mode matricization are exploited.
• Able to detect rank up to the square root of the product of all dimension lengths.
• Accuracy comparable to CP-decomposition based tensor rank detectors.

Detecting the number of components of the CANDECOMP/PARAFAC (CP) model, also known as CP model order selection, is an essential task in signal processing and data mining applications. Existing multilinear detection algorithms for handling N  -dimensional data, where N≥3N≥3, e.g., the CORe CONsistency DIAgnostic, rely on the CP decomposition which is computationally very expensive. An alternative solution is to rearrange the tensor as a matrix using the unfolding operation and then utilize the eigenvalues of the resultant matrices for CP model order selection. We propose to employ the eigenvalues associated with the unfolding along merged dimensions, namely, the multi-mode eigenvalues, as well as the n-mode eigenvalues for accurate rank detection. These multiple sets of eigenvalues are combined via the information theoretic criterion. By designing a sequential detection scheme starting from the most squared unfolded matrix, the identifiable rank is increased to the square root of the product of all dimension lengths, which renders the detection algorithm to estimate the rank that can exceed any individual dimension length. The conditions under which the proposed multilinear detection algorithm correctly detects the tensor rank are theoretically investigated and its computational efficiency and detection performance are verified.